Cyclic number (group theory)

A cyclic numberCarmichael Multiples of Odd Cyclic Numbers is a natural number n such that n and φ(n) are coprime. Here φ is Euler's totient function. An equivalent definition is that a number n is cyclic if and only if any group of order n is cyclic.See T. Szele, Über die endlichen Ordnungszahlen zu denen nur eine Gruppe gehört, Com- menj. Math. Helv., 20 (1947), 265–267.

Any prime number is clearly cyclic. All cyclic numbers are square-free.For if some prime square p2 divides n, then from the formula for φ it is clear that p is a common divisor of n and φ(n). Let , where the pi are distinct primes, then . If no pi divides any , then n and φ(n) have no common (prime) divisor, and n is cyclic.

The first non-prime cyclic numbers are 1, 15, 33, 35, 51, 65, 69, 77, 85, 87, 91, 95, 115, 119, 123, 133, 141, 143, 145, 159, 161, 177, 185, 187, 209, 213, 215, 217, 221, 235, 247, 249, 255, 259, 265, 267, 287, 295, 299, 303, 319, 321, 323, 329, 335, 339, 341, 345, 365, 371, 377, 391, 393, 395, 403, ...

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